Skip to main content

Breaking the curse of dimensionality: numerical challenges in high dimensional analysis and simulation

Final Report Summary - BREAD (Breaking the curse of dimensionality: numerical challenges in high dimensional analysis and simulation)

Many problems from science and engineering involve a large number N of variables. For this reason, their numerical treatment faces the bottleneck of the so-called "curse of dimensionality'', that refers to the exponential growth in N of the computational cost required to achieve a prescribed accuracy in the numerical result. Such computational problems are ubiquitous in statistical learning, parametric and stochastic partial differential equations (PDEs), response surfaces and parameter optimisation in numerical codes.

The first pillar of the project was of theoretical nature: identify the fundamental mathematical principles which allow to circumvent the curse of dimensionality and how they come into play in the afore mentioned applications. This was achieved for a mild class of linear and nonlinear parametrised PDEs for which sparse high-dimensional polynomial approximations can be rigorously proved to converge with rates that are immune to the curse of dimensionality.

The second pillar of the project was of numerical nature: develop computational strategy exploiting and benefiting from these mathematical principle. Our main achievement was to introduce a class of algorithms that are non-intrusive in the sense that they only require the evaluation of a finite number M of particular evaluations corresponding to well-chosen parameter instances, and which provably converge as M grows with a similar rate as predicted by the theoretical approximation results.